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Understanding Exponential Functions: Linear vs Exponential Growth, Examples, and Applications

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Explore the difference between linear and exponential functions, learn about exponential growth, decay, and applications in various fields, with practical examples and calculations provided.

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Understanding Exponential Functions: Linear vs Exponential Growth, Examples, and Applications

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  1. Exponential Functions LSP 120 Joanna Deszcz Week 2

  2. Linear vs. Exponential • Linear • Constant rate of change (slope) • y2-y1x2-x1 • Uses formula y=mx+b • Exponential • For a fixed change in x there is a fixed percent change in y • Percent change formula is new-old old • Uses formula y=P*(1+r)x

  3. Percent Change • A fixed percent change in y indicates that a function is exponential • Formula for percent change • Difference or new - old original old • For Example:

  4. Let’s Try Some Which of the following are exponential? Use the percent change formula to figure it out.

  5. Is it exponential? • If the percent change is constant, the function is exponential

  6. Exponential Function Equation • General equation is • y = P * (1 + r)x • P is initial value or value of y when x=0 • r is percent change – written as a decimal • x is input variable (usually time)

  7. Where do we use exponential functions? • Populations tend to growth exponentially not linearly • When an object cools (e.g., a pot of soup on the dinner table), the temperature decreases exponentially toward the room temperature • Radioactive substances decay exponentially • Bacteria populations grow exponentially

  8. Continued… • Money in a savings account with a fixed rate of interest increases exponentially • Viruses and even rumors tend to spread exponentially through a population (at first) • Anything that doubles, triples, halves over a certain amount of time • Anything that increases or decreases by a percent

  9. Exponential Growth? • If a quantity changes at a fixed percentage • It grows or decays exponentially

  10. Change a number by percentage • 2 ways • N = P + P * r • N= P * (1 + r) • N is ending value • P is starting value and • r is percent change • By the distributive property, equations are the same • 2nd version is preferred

  11. Percent Decrease • Same formulas except • N = P - P * r • N= P * (1 - r)

  12. Examples: • Increase 50 by 10% • N= 50 + 50 * .1 = 50 + 5 = 55 OR • N = 50 * (1+.10) = 50 * 1.10 = 55

  13. Examples cont’d • Sales tax in Chicago is 9.75%. You buy an item for $42.00. What is the final price of the article? • N = 42 + 42 * .0975 = 46.095 or • N = 42 * (1+.0975) = 46.095 • Round value to 2 decimal places since it’s money • Final answer: $46.01

  14. One More… • In 2008, the number of crimes in Chicago was 168,993. Between 2008 and 2009 the number of crimes decreased 8.7%. How many crimes were committed in 2009? • N = 168993 - 168993 * .087 = 154,290.6 or • N = 168993 * (1-.087) = 154,290.6 • Round to nearest whole number • 154,291 crimes

  15. Exponential Growth • Increase by same percent over and over • If a quantity P is growing by r % each year, • after one year there will be P*(1 + r) • after 2 years there will be P*(1 + r)2 • after 3 years there will be P*(1 + r)3 • Each year the exponent increases by one since you multiply what you already had by another (1 + r)

  16. Exponential Decay • Decrease by same percent over and over • If a quantity P is decaying by r % each year, • after one year there will be P*(1 - r) • after 2 years there will be P*(1 - r)2 • after 3 years there will be P*(1 - r)3

  17. Example • A bacteria population is at 100 and is growing by 5% per minute. • How many bacteria cells are present after one hour? • How many minutes will it take for there to be 1000 cells. • Use Excel to answer each question

  18. Where to begin? • 5% increase per minute (.05) • Multiply population by 1.05 each minute Note: no exponent needed here since filling column gives us the population each minute

  19. Don’t feel like filling? • Use the “by hand” formula with the exponent • y = P *(1+r)x • In this case: • y= 100 * (1 + .05)60 • 100 is population at start • .05 is 5% increase • 60 is number of minutes • Remember to round appropriately

  20. Radioisotope Dating One way Exponential Functions are used

  21. What is Radioactivity? • Emission of energy (or particles) from the nuclei of atoms • Atoms like to have the same # of protons and neutrons • Like to be stable • Unstable atoms are radioactive • Throw off parts to become stable

  22. Good or Bad • Good • Used in medicine to treat heart disease, cancer • Kills bacteria like salmonella • Bad • Uncontrolled nuclear chain reactions can cause major damage • Like Chernobyl • Can cause burns to cell mutations

  23. Radioisotope Dating • Uses Exponential functions • Example: • We date the Dead Sea Scrolls which have about 78% of the normally occurring amount of Carbon 14 in them.  Carbon 14 decays at a rate of about 1.202% per 100 years. • How old are the Dead Sea Scrolls?

  24. Dead Sea Scrolls • Use the formula y=P*(1-r) (measuring decay) • Fill the table until % carbon reaches about 78%

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